Multicarrier transmission has been widely adopted recently in wireline and wireless communication systems such as asymmetric digital subscriber line (ADSL) system, digital video broadcast (DVB), wireless local/metropolitan area networks (WLAN/WMAN). Exploiting discrete multitone modulation (DMT) or orthogonal frequency division multiplexing (OFDM), these systems achieve greater immunity to multipath fading and impulse noise with lower cost. However they also suffer from a high peak-to-average power ratio (PAR) problem. Without additional appropriate processing, the high PAR of a transmit signal causes a high power amplifier (HPA) to operate in its nonlinear region, which leads to spectral growth, out-of-band radiation and performance degradation.
Mathematically, the PAR for a given L times oversampled OFDM block of digital samples x can be written as:
                              PAR          =                                                    max                                  0                  ≤                  k                  ≤                                      NL                    -                    1                                                              ⁢                                                                                      x                    ⁡                                          [                                              k                        /                        L                                            ]                                                                                        2                                                    E              ⁢                              {                                                                                                x                      ⁡                                              [                                                  k                          /                          L                                                ]                                                                                                  2                                }                                                    ⁢                                  ⁢                                            where              ⁢                                                          ⁢                              x                ⁡                                  [                                      k                    /                    L                                    ]                                                      =                                          x                ⁡                                  (                                      k                    ·                                          T                      /                      L                                                        )                                            +                                                1                  N                                ⁢                                                      ∑                                          n                      =                      0                                                              N                      -                      1                                                        ⁢                                                            X                      n                                        ·                                          ⅇ                                              j2π                        ⁢                                                                                                  ⁢                                                  kn                          /                          NL                                                                                                                                                  ,                                          ⁢                      k            =            0                    ,          1          ,          …          ⁢                                          ,                      NL            -            1                                              (        1        )            Xn, n=0, 1, . . . , N−1 is the data symbol modulated onto the nth subcarriers, 1/NT is the subchannel spacing. E{ } denotes the expected value.
Tone reservation is one of the important PAR reduction techniques. It modulates unavailable or reserved tones to produce a data-block-dependent peak canceling signal c, so that the maximum magnitude of the output signal x+c=IDFT(X+C) is below the desired peak power level and the PAR of the output signal is lower than that of x. The vectors X=[X0, X1, . . . , XN-1] and C=[C0, C1, . . . , CN-1] cannot both be nonzero on a given subcarrier:
                                          X            k                    +                      C            k                          =                  {                                                                                          X                    k                                    ,                                      k                    ∈                    U                                                                                                                                            C                    k                                    ,                                      k                    ∈                                          U                      c                                                                                                                              (        2        )            where N subcarriers in an multicarrier system are divided into two subsets: the subcarrier set U for useful data and the subcarrier set Uc for symbols optimized to reduce PAR. The key problem in tone reservation is how to produce the peak canceling signal with low computation cost.
An active-set approach is a well-known and efficient way to resolve the linear optimization problem, which can be applied directly to design an optimum peak canceling signal.
In the prior art, a peak-reduction kernel p0 is used as the basis of a PAR-reduction signal for the algorithm, which is computed by projecting an impulse at location n=0 onto the set of reserved tones and is scaled to have unit value at n=0. The active set method is summarized as follows:    1) Begin with x0=x, Set i=1, and Let E0 be the maximum magnitude. The active set contains the maximum-magnitude sample at location n1.    2) Set p1=pn1, pn1 is obtained by circularly shifting p0 at n1.    3) Perform peak-testing with xi-1 and pi.    4) Find the minimum step size μi with equation (3) and compute Ei=Ei-1−μi. Add the peak associated with μi to the active set.
                              μ          i                =                              min                          q              ∉              A                                ⁢                      (                                                                                E                                          i                      -                      1                                                        -                                                                                x                      q                                              i                        -                        1                                                                                                                                  1                  -                                                            sgn                      ⁡                                              (                                                  x                          q                                                      i                            -                            1                                                                          )                                                              ⁢                                          p                      q                      i                                                                                  ≥              0                        )                                              (        3        )                5) Compute xi=xi-1−μipi     6) If a maximum number of iteration is reached, or the desired peak-power level W is reached, then STOP.    7) Set up the matrix equation (4) and solve for αs.
                                          [                                                            1                                                                      p                                                                  n                        1                                            -                                              n                        2                                                                                                              …                                                                      p                                                                  n                        1                                            -                                              n                        i                                                                                                                                                              p                                                                  n                        2                                            -                                              n                        1                                                                                                              1                                                  …                                                                      p                                                                  n                        2                                            -                                              n                        i                                                                                                                                          ⋮                                                                                                                                                                                                                                  ⋮                                                                                                  p                                                                  n                        i                                            -                                              n                        1                                                                                                                                  p                                                                  n                        i                                            -                                              n                        1                                                                                                              …                                                  1                                                      ]                    ⁡                      [                                                                                α                    1                                                                                                                    α                    2                                                                                                ⋮                                                                                                  α                    i                                                                        ]                          =                  [                                                                      S                                      n                    1                                                                                                                        S                                      n                    2                                                                                                      ⋮                                                                                      S                                      n                    i                                                                                ]                                    (        4        )            
Where pn is the n th entry of p0 and Sni=sign(xnii-1)
      p    i    =            ∑              l        =        1            i        ⁢                  α        l            ⁢              p                  n          l                        9) Go to STEP 3.
The prior art method increases the peaks in the active set by one and calculates the whole peak canceling signal for these peaks in each iteration in order to get the optimum PAR reduction performance. But in practice, what is needed is not the maximum PAR reduction, but reducing the PAR to the desired range or reducing the maximum magnitude below the desired peak power level. In the prior art method, calculating the whole peak canceling signal and balancing the whole output signal in each iteration, which increases delay and computation cost greatly, is not necessary.
What is desired is an approach that enables comparable PAR reduction, but with less computation requirement and much less maximum delay.